Results about existence of a signed ground state solution and multiple solutions (if $f$ is odd with respect to the second variable) are proven for a class of asymptotically linear elliptic problems involving a Carathéodory type nonlinearity satisfying assumptions weaker than (fF) in \cite{GLZ} or $(F_{2})_{+}$ in \cite{CM}. A close relation between the behaviour of $\lim_{|t|\to \infty}f(x, t)/|t|$ and the number of solutions is stablished.
We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.
We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.
We show that resonant states in scattering on asymptotically hyperbolic man-ifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.
We consider a double phase Robin problem with a Carathéodory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti-Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
In this paper we prove the existence of a signed ground state solution in the mountain pass level for a class of asymptotically linear elliptic problems, even when the nonlinearity is just continuous in the second variable. The (strongly) resonant and non-resonant cases are discussed. A multiplicity result is also proved when $f$ is odd with respect to the second variable.
We consider a nonlinear Robin problems driven by the $p$-Laplacian plus an indefinite potential. The reaction is resonant with respect to a variational eigenvalue. For the principal eigenvalue we assume strong resonance. Using variational tools and critical groups we prove existence and multiplicity theorems.
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for…
The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which is just continuous. Also in the case when the problem has not a variational structure, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is "stable" under small perturbations, obtaining multiplicity results if the nonlinearity is odd, both in the non--reso…
We study the following fractional elliptic equations of the type, \begin{equation*} (-Δ)^{\frac12}_A u = λu+f(|u|)u ,\;\textrm{in } \;(-1, 1),\; u=0\;\textrm{in } \;\mathbb R\setminus (-1, 1), \end{equation*} where $λ$ is a positive real parameter and $(-Δ)^{\frac12}_A$ is the fractional magnetic operator with $A:\mathbb R\to \mathbb R$ being a smooth magnetic field. Using a classical critical point theorems, we prove the existence of multiple solutions in the non-resonant case when the nonlinear term $f(t)$ has a critical exponential growth in the sense of Trudinger-Moser inequality.
We consider an asymptotically linear Schrödinger equation $-Δu + V(x)u = λu + f(x,u), \ x\in R^N$, and show that if $λ_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence $(u_n,λ_n)$ of solutions such that $\|u_n\|\to\infty$ and $λ_n\toλ_0$. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of $λ_0$ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.
We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equatio…
We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the the pseudo-index theory developed by Bartolo, Benci and Fortunato \cite{bbf} after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in literature, some new additional d…
We establish existence and multiplicity theorems for a Dirichlet boundary value problem at resonance, which is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta. Our framework includes a sign-changing potential and we locate the solutions by using the Mountain Pass lemma and the Saddle Point theorem.
We prove existence and multiplicity of small amplitude periodic solutions of the completely resonant wave equation u_tt - u_xx + f(x,u)=0 with Dirichlet boundary conditions where f(x,u)=a_2 u^2 + a_3(x) u^3 + O(u^4) or f(x,u)= a_4 u^4 + O(u^5) for a Cantor-like set of frequencies omega of asymptotically full measure at omega=1.
We study a nonlinear boundary value problem driven by the $p$-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carathéodory function which is asymptotically resonant at $\pm\infty$ with respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign.
The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem \[ \left\{\begin{array}{lr} - \divg (A(x,u)\, |\nabla u|^{p-2}\, \nabla u) + \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\ f(x,u) & \hbox{in $Ω$,}\\ u\ = \ 0 & \hbox{on $\partialΩ$,} \end{array} \right. \] where $Ω\subset \R^N$ is a bounded domain, $N\ge 2$, $p > 1$, $A$ is a given function which admits partial derivative $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $f$ is asymptotically $p$-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if $A(…
A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.
A unified treatment of degenerate nonlocal elliptic problems
University of the National Education Commission（国家教育委员会大学）；CIEM-Conicet, Universidad Nacional de Córdoba（科尔多瓦国立大学CONICET-CEM研究所）；Universidade Federal do Pará（帕拉联邦大学）；Universidade Federal de Goiás（戈亚斯联邦大学）
We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -Δu + λu - μ\frac{u}{|x|^2} - ν\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus Ω)^2} = f(x,u) & \quad \mbox{in } Ω\\ u = 0 & \quad \mbox{on } \partial Ω, \end{array} \right. $$ on a bounded domain $Ω\subset \mathbb{R}^N$ with $0 \in Ω$. We assume that the nonlinear part is superlinear on some closed subset $K \subset Ω$ and asymptotically linear on $Ω\setminus K$. We find a solution with the energy bounded by a certain min-max level, and infinitely many solutions provided that $f$ is odd in $u$. Moreover we stu…
We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global opt…
In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions for a $2\times2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern--Simons vortices. We show that the given system admits at least two solutions when the Chern--Simons coupling parameter $κ>0$ is sufficiently small; while no solutions exist for $κ>0$ sufficiently large. As in [36] we use a variational formulation of the problem. Thus, we obtain a first solution via a (local) minimization method and show that it is asymptotically gauge-equivalent to the (broken) pri…
